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Topic: Euler-Maclaurin summation formula

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Euler-Maclaurin formula - Wikipedia, the free encyclopedia The Euler-MacLaurin summation formula can thus be seen to be an outcome of the representation of functions on the unit interval by the direct product of the Bernoulli polynomials and their duals. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals. In particular, sin(2πnx) lies in the kernel; the integral of sin(2πnx) is vanishing on the unit interval, as is the difference of its derivatives at the endpoints. en.wikipedia.org /wiki/Euler-Maclaurin_formula

Formula Small-angle formula The small-angle formula is a angular size (α) and its distance from the observer (d) and is a... Trinitarian formula The trinitarian formula is the phrase "in the name of the Father, and of the Son, and of the Holy Sp... Formula editor A formula editor is a name for a computer program that is used to typeset mathematical works or formulae.... www.brainyencyclopedia.com /topics/formula.html

Trapezium rule - Wikipedia, the free encyclopedia Moreover, the trapezium rule tends to become extremely accurate when periodic functions are integrated over their periods, a fact best understood in connection with the Euler-Maclaurin summation formula. The trapezium rule is one of a family of formulas for numerical integration called Newton-Cotes formulas. Simpson's rule and other like methods can be expected to improve on the trapezium rule for functions which are twice continuously differentiable; however for rougher functions the trapezium rule is likely to prove preferable. en.wikipedia.org /wiki/Trapezium_rule

gamma.html Euler-Maclaurin summation was used with n=10 [ 2 ]. The Euler-Maclaurin summation can be used to have a complete asymptotic expansion of the harmonic numbers. Integral and series formulae for the Euler constant can be found in Collection of formulae for the Euler constant. numbers.computation.free.fr /Constants/Gamma/gamma.html

References for Maclaurin J Mooney, Colin Maclaurin and Glendaruel, The Mathematical Intelligencer 16 (1) (1994), 48-49. H W Turnbull, Bi-centenary of the death of Colin Maclaurin (1698-1746), mathematician and philosopher, professor of mathematics in Marischal College, Aberdeen (1717-1725) (Aberdeen, 1951). The continental influence of Maclaurin's treatise of fluxions, Amer. www-history.mcs.st-and.ac.uk /history/Printref/Maclaurin.html

Boundary conditions for Euler Maclaurin summation formula Here is a link to the article on the Euler-Maclaurin Summation formula on Mathworld, http://mathworld.wolfram.com/Euler-MaclaurinIntegrationFormulas.html The phrase of interest to me is: In certain cases, the last term tends to 0 as n tends to infinity. Here is a link to the article on the =Euler-Maclaurin Summation formula on Mathworld, =http://mathworld.wolfram.com/Euler-MaclaurinIntegrationFormulas.html = =The phrase of interest to me is: In certain cases, the last term tends =to 0 as n tends to infinity. The Euler MacLaurin summation formula links the values of a sum to the value of the definite integral. www.forum-one.org /new-3627078-4348.html

PlanetMath: proof of Euler-Maclaurin summation formula This is version 2 of proof of Euler-Maclaurin summation formula, born on 2003-02-22, modified 2003-02-22. "proof of Euler-Maclaurin summation formula" is owned by pbruin. Substituting this and absorbing the left term into the summation yields Eq. planetmath.org /encyclopedia/ProofOfEulerMaclaurinSummationFormula.html

DIFFERENCES, CALCULUS OF (Theory of Finite Differences) - Online Information article about DIFFERENCES, CALCULUS OF (Theory of Finite Differences) Summation of Series.If u, denotes the (r+1)th term of a series, and if v, is a function of r such that Av,=u, for all integral values of r, then the sum of the terms u,,, u,n+1 un is (iv.) There are variants of these formulae, due to taking hum+4 as the first approximation to the area of the curve between um and um+i; the formulae involve the sum u4+u4+. which are identical with the formulae in (ii.) and (i.) of 3. encyclopedia.jrank.org /DEM_DIO/DIFFERENCES_CALCULUS_OF_Theory.html

Maclaurin He had what is now known as the Euler-Maclaurin summation formula, a beautiful formula relating the sum of series to integrals and having many applications to numerical analysis, a year or two before Euler also discovered it. Maclaurin was a child prodigy who began his studies at the University of Glasgow at age eleven; at age fourteen he received his master's degree. Nowadays Maclaurin is best known for something he never claimed to have invented, the Maclaurin series which is just the Taylor series centered at the origin. www.math.fau.edu /schonbek/Modern_Analysis/calcmath12.html

MCS 471 Lecture Thirty One The justification of Romberg integration involved the Euler Maclaurin summation formula. A note on the Euler-Maclaurin summation formula is available The formula for Richardson extrapolation with central differences works because the central difference formula is an even function in h. www.math.uic.edu /~jan/mcs471/Lec31/lec31.html

The Euler-Maclaurin formula for simple integral polytopes -- Karshon et al. 100 (2): 426 -- Proceedings of the National Academy of Sciences The classical Euler-Maclaurin summation formula with remainder for a function f of class C The early references to the Euler-Maclaurin formula are Euler ( 3, 4) and Maclaurin ( 5), although apparently Poisson formula is a generalization of the exact version of Eq. www.pnas.org /cgi/content/full/100/2/426

Bernoulli Bibliography: S [1] On unification and extension of Bernoulli, Euler and Eulerian polynomials. [1] An extension of the Bernoulli and Euler polynomials, Jñanabha, 20 (1990), 7-12. [1] On the extension of Bernoulli, Euler and Eulerian polynomials. www.mscs.dal.ca /~dilcher/berns.html

monograf.html Substitute the Euler-Maclaurin summation formula in the limit part of Euler's constant. This is a recursive formula to generate Euler numbers. This is a recurrence formula for the gamma function. www.getnet.com /~cherry/mathml/monograf.html

10.2 The Euler-Maclaurin Summation Formula It is clear that the integral alone (0.0215108) is not a very good representation of the sum, but that the Euler-Maclaurin summation formula does exceptionally well in estimating the sum. A standard method for doing this is called the Euler-Maclaurin summation formula. To use the Euler-Maclaurin formula we will need: scholar.chem.nyu.edu /2600/classnotes/node106.html

Lesson 28 Euler-Maclaurin Summation Formula # --------------------------------------------- # > restart: # Suppose f is a smooth function, and F an anti-difference of f, i.e. ----- k = 1 # Basically what we want to do is "invert" this formula to write F(x) in terms of # f(x), its antiderivative and its derivatives. \ k = 1 / # So our formula for f becomes > subs(", ""); infinity ----- (k) \ D (F)(x) f(x) =) ---------- / k! www.ugrad.math.ubc.ca /coursedoc/m210/lesson28.html

Open Directory - Science: Math: Reference Summation Formulae - For polynomials of degrees up to 50, generated using the Euler-Maclaurin method. Geometry Formulas and Facts - An excerpt from the 30th Edition of the CRC Standard Mathematical Tables and Formulas, covering the area of Geometry (minus differential geometry), by Silvio Levy. Math Assistance - Lists rules and formulas for a number of mathematical subjects, such as plotting graphics, functions, factoring, derivatives, integrals, matrices, vectors, and numerical analysis. dmoz.org /Science/Math/Reference

Recently archived material [abstract:] "We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpinski gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial. It is proved that this zeta function has a meromorphic continuation to a half plain with poles contained in an arithmetic progression. www.maths.ex.ac.uk /~mwatkins/zeta/newmaterial.htm

Hybrid Gauss-Trapezoidal Quadrature Rules The new nodes and weights are determined so that the asymptotic expansion of the resulting rule, provided by a generalization of the Euler--Maclaurin summation formula, has a prescribed number of vanishing terms. Euler--Maclaurin formula, Gaussian quadrature, high-order convergence, numerical integration, positive weights, singularity The quadratures result from alterations to the trapezoidal rule, in which a small number of nodes and weights at the ends of the integration interval are replaced. epubs.siam.org /sam-bin/dbq/article/32514

Mathematics Magazine: Euler-Maclaurin and Taylor formulas: Twin, elementary derivations, The In fact, neither Euler nor Maclaurin found this formula with remainder; the first to do so was Poisson, in 1823 ([14], see also [8, p. The E-M summation formula is among the most remarkable formulas of mathematics [15, p. Since then the E-M formula has been derived in different ways; one of the earliest derivations (1834) was presented by Jacobi [10]. www.findarticles.com /p/articles/mi_qa3789/is_200104/ai_n8950214

Introduction on Bernoulli's numbers Perhaps one of the most important result is Euler-Maclaurin summation formula, where Bernoulli's numbers are contained and which allows to accelerate the computation of slow converging series (see the essay on Euler's constant at [ 9 ]). Also in 1631, Johann Faulhaber (1580-1635) developed explicit formulas for these sums up to p=17 (read the excellent [ 12 ] for the beginnings of integration and [ 18 ] for some excerpts of Bernoulli's work). In 1735, the solution of the Basel problem, expressed by Jakob Bernoulli some years before, was one of Euler's most sensational discovery. numbers.computation.free.fr /Constants/Miscellaneous/bernoulli.html

PROBLEM SET 8 The Euler-Maclaurin summation formula is useful in a number of problems of this type. The Euler-Maclaurin formula gives a systematic way of relating the sum to the integral, hence of calculating the corrections to the classical result. The calculation here shows that Mulholland's formula gives a good approximation for the rotational partition function when the series in even and odd values of the orbital angular momentum l appear with equal weights. www.hep.wisc.edu /~ldurand/715html/homework/set8.html

Summation formulas I generated those formulas using Euler-Maclaurin summation method: After many searches I couldn't find a single page containing summation formulas for polynomials of order greater than 4. Fixed typo in E.M. summation formula, added descript. polysum.tripod.com

The Euler-Maclaurin formula revisited The Euler-Maclaurin summation formula for the approximate evaluation of I = \int anziamj.austms.org.au /V40/E005/home.html

10.3 The Evaluation of the Partition Function To use the Euler-Maclaurin summation formula we need some derivatives. Thus the 1 in the Euler-Maclaurin summation is conveniently forgotten. To this end it helps to gather all the constants together in one bunch and call them, say, a : scholar.chem.nyu.edu /2600/classnotes/node107.html

Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations In the absence of a singularity, this error in described by the classical Euler-Maclaurin summation formula, which is an asymptotic expansion in inverse integer powers of m. The context of this note in the discretization error made by the m-panel trapezoidal rule when the integrand has an algebraic singularity at one end, say x = 0, of the finite integration interval. www.osti.gov /energycitations/product.biblio.jsp?osti_id=10121989

Bernoulli Bibliography: B BATEN W. [1] A remainder for the Euler-Maclaurin summation formula in two independent variables, Amer. [1] Periodic analogues of the Euler-Maclaurin and Poisson summation formulas with applications to number theory, Acta Arith., 28 (1975), 23-68. [1] The generalised Euler formula from Poisson's summation formula and some applications, J. Phys. www.mscs.dal.ca /~dilcher/bernb.html

E25 -- Methodus generalis summandi progressiones Mills S., “The independent derivations by Euler, Leonhard and Maclaurin, Colin of the Euler-Maclaurin summation formula.” Archive for History of Exact Sciences, 33 (1-3), pp. Ferraro G., “Some aspects of Euler's theory of series: Inexplicable functions and the Euler-Maclaurin summation formula.” Historia Mathematica, 25 (3), pp. Dutka J., “On the summation of some divergent series of Euler and the zeta functions.” Archive for History of Exact Sciences, 50 (2), pp. math.dartmouth.edu /~euler/pages/E025.html

Entry Lyness:1985:EME from nummath.bib Lab., IL, USA", keywords = "approximation; asymptotic expansion; Cauchy principal value integral; Euler Maclaurin expansion; Euler Maclaurin summation formula; Fourier coefficient asymptotic expansion; function approximation; integration; numerical quadrature; trapezoidal rule", treatment = "T Theoretical or Mathematical", } summation, 7(0) 147, 21(1) 81, 22(5) 393, 48(2) 199, 51(1) 37, 58(6) 583, 80(1) 61 Lyness, J. Maclaurin, 15(4) 333, 27(3) 355, 55(3) 281 www.math.utah.edu /ftp/pub/tex/bib/idx/nummath/46/4/611-622.html

Untitled Excerpts on the Euler-Maclaurin summation formula, from Institutiones Calculi Differentialis by Leonhard Euler (pdf format) Representations of Catalan's constant, (an unpublished catalog of formulae for the alternating sum of the reciprocals of the odd positive squares), 1998. Borwein and Bradley's Ap 'ery-Like Formulae for i (4n + 3) Gert... www.geocities.com /dahlak10021/math.htm

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